Step 1
The confidence interval for a sample proportion at \(\displaystyle{95}\%\) can be calculated using the formula,
\(\displaystyle{w}{i}{d}{e}\hat{{{p}}}-{z}\cdot\sqrt{{{\frac{{{w}{i}{d}{e}\hat{{{p}}}{\left({1}-{w}{i}{d}{e}\hat{{{p}}}\right)}}}{{{N}}}}}}{<}{p}{<}{w}{i}{d}{e}\hat{{{p}}}+{z}\cdot\sqrt{{{\frac{{{w}{i}{d}{e}\hat{{{p}}}{\left({1}-{w}{i}{d}{e}\hat{{{p}}}\right)}}}{{{N}}}}}}\)</span>, where \(\displaystyle{w}{i}{d}{e}\hat{{{p}}}\) is the sample proportion, N is the sample size and z* is the approprite value from the standard normal distributions for the desired confidence intervals.
The z* value for \(\displaystyle{95}\%\) confidence interval is equal to 1.96
Step 2
The sample proportion is equal to 0.66 and the sample size is equal to 110 and the z* value is equal to 1.96
Substitute the values of \(\displaystyle{w}{i}{d}{e}\hat{{{p}}}\), N and z* in the formula, \(\displaystyle{w}{i}{d}{e}\hat{{{p}}}-{z}\cdot\sqrt{{{\frac{{{w}{i}{d}{e}\hat{{{p}}}{\left({1}-{w}{i}{d}{e}\hat{{{p}}}\right)}}}{{{N}}}}}}{<}{p}{<}{w}{i}{d}{e}\hat{{{p}}}+{z}\cdot\sqrt{{{\frac{{{w}{i}{d}{e}\hat{{{p}}}{\left({1}-{w}{i}{d}{e}\hat{{{p}}}\right)}}}{{{N}}}}}}\)</span> to calculate the confidence interval of the sample proportion at \(\displaystyle{95}\%\) interval
\(\displaystyle{w}{i}{d}{e}\hat{{{p}}}-{z}\cdot\sqrt{{{\frac{{{w}{i}{d}{e}\hat{{{p}}}{\left({1}-{w}{i}{d}{e}\hat{{{p}}}\right)}}}{{{N}}}}}}{<}{p}{<}{w}{i}{d}{e}\hat{{{p}}}+{z}\cdot\sqrt{{{\frac{{{w}{i}{d}{e}\hat{{{p}}}{\left({1}-{w}{i}{d}{e}\hat{{{p}}}\right)}}}{{{N}}}}}}\)</span>

\(\displaystyle{0.66}-{\left({1.96}\sqrt{{{\frac{{{0.66}{\left({1}-{0.66}\right)}}}{{{110}}}}}}\right)}{<}{p}{<}{0.66}+{\left({1.96}\sqrt{{{\frac{{{0.66}{\left({1}-{0.66}\right)}}}{{{110}}}}}}\right)}\)</span>

\(\displaystyle{0.66}-{\left({1.96}\sqrt{{{0.00204}}}\right)}{<}{p}{<}{0.66}+{\left({1.96}\sqrt{{{0.00204}}}\right)}\)</span>

\(\displaystyle{0.66}-{\left({1.96}{\left({0.0451}\right)}{<}{p}{<}{0.66}+{\left({1.96}{\left({0.0451}\right)}\right.}\right.}\)</span>

\(\displaystyle{0.66}-{0.0884}{<}{p}{<}{0.66}+{0.0884}\)</span>

\(\displaystyle{0.5716}{<}{p}{<}{0.7484}\)</span> The confidence interval for the sample proportion at \(\displaystyle{95}\%\) is equal to (0.5716, 0.7484)

\(\displaystyle{0.66}-{\left({1.96}\sqrt{{{\frac{{{0.66}{\left({1}-{0.66}\right)}}}{{{110}}}}}}\right)}{<}{p}{<}{0.66}+{\left({1.96}\sqrt{{{\frac{{{0.66}{\left({1}-{0.66}\right)}}}{{{110}}}}}}\right)}\)</span>

\(\displaystyle{0.66}-{\left({1.96}\sqrt{{{0.00204}}}\right)}{<}{p}{<}{0.66}+{\left({1.96}\sqrt{{{0.00204}}}\right)}\)</span>

\(\displaystyle{0.66}-{\left({1.96}{\left({0.0451}\right)}{<}{p}{<}{0.66}+{\left({1.96}{\left({0.0451}\right)}\right.}\right.}\)</span>

\(\displaystyle{0.66}-{0.0884}{<}{p}{<}{0.66}+{0.0884}\)</span>

\(\displaystyle{0.5716}{<}{p}{<}{0.7484}\)</span> The confidence interval for the sample proportion at \(\displaystyle{95}\%\) is equal to (0.5716, 0.7484)